Page 240 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes M = (M')'. If M is not a suitable N, we form M = M )', and so on. The point is that the strict
(2)
(3)
(2)
(2)
inequalities
(2)
l(M ) > l(M’ ) > l(M > ...
i 1 1
cannot continue for very long. After not more than l(M ) iterations of our procedure, we must
1
arrive at a matrix M which has the properties we seek.
(k)
Theorem 1: Let P be an m m matrix with entries in the polynomial algebra F[x]. The following
are equivalent.
(i) P is invertible.
(ii) The determinant of P is a non-zero scalar polynomial.
(iii) P is row-equivalent to the m m identity matrix.
(iv) P is a product of elementary matrices.
Proof: Certainly (i) implies (ii) because the determinant function is multiplicative and the only
polynomials invertible in F[x] are the non-zero scalar ones. Our argument here provides a proof
that (i) follows from (ii). We shall complete the merry-go-round
(i) (ii)

(iv) (iii).

The only implication which is not obvious is that (iii) follows from (ii).
Assume (ii) and consider the first column of P. It contains certain polynomials p , ... , p , and
1 m
g.c.d. (p , ..., p ) = 1
1 m
because any common divisor of p , ..., p . must divide (the scalar) det P. Apply the previous
1 m
lemma to P to obtain a matrix
1 a 2  a m
0
Q = ...(5)
 B
0

which is row-equivalent to P. An elementary row operation changes the determinant of a matrix
by (at most) a non-zero scalar factor. Thus det Q is a non-zero scalar polynomial. Evidently the
(m – 1) (m – 1) matrix B in ( 5) has the same determinant as does Q. Therefore, we may apply the
last lemma to B. If we continue this way for m steps, we obtain an upper-triangular matrix
1 a 2  a m
0 1  b m
R =
  
0 0  1

which is row-equivalent to R. Obviously R is row-equivalent to the m m identity matrix.
Corollary: Let M and N be m n matrices with entries in the polynomial algebra F]x]. Then N is
row-equivalent to M if and only if

N = PM
where P is an invertible m m matrix with entries in F[x].

234 LOVELY PROFESSIONAL UNIVERSITY